A Navier-Stokes phase-field crystal model for colloidal suspensionsSimon Praetorius and Axel Voigt Citation: The Journal of Chemical Physics 142, 154904 (2015); doi: 10.1063/1.4918559 View online: http://dx.doi.org/10.1063/1.4918559 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/15?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Parabolized Navier–Stokes model for study the interaction between roughness structures and concentratedvortices Phys. Fluids 25, 104103 (2013); 10.1063/1.4823746 Short-time transport properties in dense suspensions: From neutral to charge-stabilized colloidal spheres J. Chem. Phys. 128, 104903 (2008); 10.1063/1.2868773 A smooth particle-mesh Ewald algorithm for Stokes suspension simulations: The sedimentation of fibers Phys. Fluids 17, 033301 (2005); 10.1063/1.1862262 Sedimentation equilibrium of a suspension of adhesive colloidal particles in a planar slit: A density functionalapproach J. Chem. Phys. 116, 384 (2002); 10.1063/1.1421354 Sedimentation potential of a concentrated spherical colloidal suspension J. Chem. Phys. 110, 11643 (1999); 10.1063/1.479103

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THE JOURNAL OF CHEMICAL PHYSICS 142, 154904 (2015)

A Navier-Stokes phase-field crystal model for colloidal suspensionsSimon Praetoriusa) and Axel Voigtb)

Institute of Scientific Computing, Technische Universität Dresden, D-01062 Dresden, Germany

(Received 26 January 2015; accepted 3 April 2015; published online 20 April 2015)

We develop a fully continuous model for colloidal suspensions with hydrodynamic interactions.The Navier-Stokes Phase-Field Crystal model combines ideas of dynamic density functional theorywith particulate flow approaches and is derived in detail and related to other dynamic densityfunctional theory approaches with hydrodynamic interactions. The derived system is numericallysolved using adaptive finite elements and is used to analyze colloidal crystallization in flow-ing environments demonstrating a strong coupling in both directions between the crystal shapeand the flow field. We further validate the model against other computational approaches forparticulate flow systems for various colloidal sedimentation problems. C 2015 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4918559]

I. INTRODUCTION

Simple fluids can be coarse grained, considered as a con-tinuum and very well described by the Navier-Stokes equa-tions. A quantitative description can be achieved down to thenanometer scale. For colloidal suspensions, this simple treat-ment is not necessarily valid any more. Here, colloidal parti-cles, with a typical size range of nanometers to a few microns,move due to collisions with the solvent molecules, interact witheach other and induce flow fields due to their motion. It isshown that these hydrodynamic interactions are of relevance invarious practical applications, e.g., colloidal gelation1 or coag-ulation of colloidal dispersions.2 To calculate nonequilibriumproperties of such systems, it requires to resolve the differenttime and length scales arising from thermal Brownian motionand hydrodynamic interactions. Various approaches have beendeveloped to consider these interactions in an effective way.For an overview and proposed coarse-graining descriptions,see e.g., Ref. 3.

One of the most popular approaches is Stokesian dy-namics (SD) within the low Reynolds number limit.4 Thehydrodynamic interaction is thereby incorporated in an approx-imate analytical form, assuming to result as the sum of two-body interactions. The approach is difficult to implementfor complex boundary conditions and is relatively expensive.As an alternative, direct numerical simulations are proposed,which involve determining fluid motion simultaneously withparticle motion. In these methods, the colloidal particles arefully resolved and coupled with the Navier-Stokes equations,leading to coupled discrete-continuous descriptions. Otherdiscrete and hybrid models are, e.g., the (smoothed) dissipativeparticle dynamics model,5–7 the fictitious domain/immersedboundary method,8,9 and the Lattice Boltzmann-moleculardynamics method.10–12 A short review and comparison of suchmodels is given in Ref. 13.

Our aim is to derive a fully continuous system of equationsfrom such hybrid models. This has the advantage of an efficient

a)simon.praetorius@tu-dresden.deb)axel.voigt@tu-dresden.de

numerical treatment, the possibility of a detailed numericalanalysis, and it offers a straight forward coupling with otherfields. The model will serve as a general continuum model forcolloidal suspension, providing a quantitative approach downto the length scale set by the colloidal particles and it is oper-ating on diffusive time scales. The approach will be derivedby combining ideas from: (a) dynamic density functionaltheory (DDFT) and (b) hybrid discrete-continuous particulateflow models. We will test the derived system for colloidalcrystallization in flowing environments and for colloidal sedi-mentation.

A. Dynamic density functional theory approach

The aim of the DDFT approach is to provide a reducedmodel that describes the local state of a colloidal fluid by thetime averaged one-particle density. The evolution of this den-sity is driven by a gradient-flow of the equilibrium Helmholtzfree-energy functional. A first realization of a DDFT forcolloidal fluids is presented in the work of Marconi and Tara-zona14 with colloids modeled as Brownian particles. Later, thistheory has been extended by Archer15 and has been connectedto the equations of motion from continuum fluid mechanics.Rauscher16 described an advected DDFT, to model colloids ina flowing environment, that do not interact via hydrodynamicinteractions. The work of Goddard et al.17 incorporates theeffect of inertia and hydrodynamic interactions between thecolloidal particles, and recently Gránásy et al.18 have exploreda coarse-grained density coupling of DDFT and the Navier-Stokes equations.

We start the derivation of our model with the dynamicalequations derived by Archer.15 Therefore, we introduce theone-particle (number) density ϱ(r, t) and the average localvelocity v(r, t) of the colloidal particles. The density is drivenby a continuity equation

∂tϱ + ∇ · (ϱv) = 0, (1)

with the current, expressed as ϱv, evolving by the dynamicalequation

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154904-2 S. Praetorius and A. Voigt J. Chem. Phys. 142, 154904 (2015)

mϱ (∂tv + (v · ∇)v + γv) = −ϱ∇ δFH[ϱ]δϱ

+ η∆v, (2)

where m represents the mass of the particles, γ a dumpingcoefficient,FH[ϱ] the equilibrium Helmholtz free-energy func-tional, and η a viscosity coefficient.

We use a minimal expression for the free-energy, theSwift-Hohenberg (SH) energy,19,20 in a dimensionless form

FH[ϱ(ψ)] ≃ Fsh[ψ] =

14ψ4 +

12ψ(r + (q2

0 + ∆)2)ψ dr,

(3)

with ϱ = ϱ(1 + (ψ + 0.5)), a parametrization of the one-parti-cle density with respect to a reference density ϱ. The phenom-enological parameter r is related to the undercooling of thesystem and the constant q0 is related to the lattice spacing.This functional arises by splitting the energy in an ideal gascontribution and an excess free energy FH = Fid + Fexc, rescal-ing and shifting of the order-parameter ϱ, expanding idealgas contributions in real-space, and the excess free energyin Fourier-space and simplification by removing constant andlinear terms that would vanish in the dynamical equations. Adetailed derivation of the energy can be found in Refs. 21–23.

Inserting the density expansion and free-energy (3) into(1) and (2), we get a system of dynamic equations for the den-sity deviation ψ and the related non-dimensionalized averagedvelocity v,

∂tψ + ∇ ·(1.5 + ψ)v

= 0, (4)

(1.5 + ψ) (∂tv + (v · ∇)v + Γv)

=1

Re∆v − 1

Pe(1.5 + ψ)∇ δFsh[ψ]

δψ. (5)

With respect to a length scale L and time scale L/V0, we havethe dimensionless variable v = v/V0, and Peclet number Pe,Reynolds number Re, and friction coefficient Γ given by

Pe =3mV 2

0

kBT, Re =

m ϱLV0

η, Γ =

γLV0,

with Boltzmann’s constant kB and temperature T . In Appen-dix A, a detailed derivation of this dimensionless form of thedynamical equations can be found.

In the overdamped limit, Γ ≫ 1, the velocity equationreduces to an explicit expression that relates the velocity to thechemical potential by

(1.5 + ψ)v = − 1ΓPe

(1.5 + ψ)∇ δFsh[ψ]δψ

. (6)

Inserting (6) into (4) results in the Phase-Field Crystal (PFC)equation20

∂tψ =1ΓPe∇ ·

((1.5 + ψ)∇ δFsh[ψ]

δψ

), (7)

referred to as PFC1 model in Ref. 21.

B. Particulate flows

Typical approaches to simulate particulate flows on largerlength scales consider a Newton-Euler equation for each par-ticle to describe their motion as a rigid body and combine

this with a Navier-Stokes solver for the flow around theseparticles. Various numerical approaches have been proposedto model this flow and the incorporation of a no-slip bound-ary condition on the particle surfaces, see, e.g., Refs. 24–27.Examples for numerical approaches are the fictitious domainand the immersed boundary method. All these approaches usethe general idea to consider the particles as a highly viscousfluid, which allows the flow computation to be done on a fixedspace region. The no-slip boundary condition on the particlesurface is thereby enforced directly or implicitly, depending onthe numerical approach. All these methods combine a contin-uous description of the flow field with a discrete off-latticesimulation for the particles.

Considering an incompressible fluid with viscosity η f

and constant fluid density ρ f , we can write the Navier-Stokesequations for velocity u and pressure p of a pure fluid in adimensionless form,

∂tu + (u · ∇)u = −∇p +1

Re f∇ ·

2(1 + η)D(u) + F, (8)

∇ · u = 0, (9)

with length and time scale as above and the dimensionlessvelocity field u = u/V0, the viscosity perturbation η from theexpansion η f = η f (1 + η), and the fluid Reynolds number Re f

and the dimensionless pressure p given by

Re f =ρ f LV0

η f, p =

pρ fV 2

0

,

respectively. The expression D(u) gives the symmetric part ofthe velocity gradient, i.e.,

D(u) = 12(∇u + ∇u⊤)

and F defines a volume force.As a reference model for colloidal suspensions, we con-

sider the fluid particle dynamics (FPD) model by Ref. 28. Here,the particles are considered as a highly viscous fluid and thevelocities of the particles are extracted from the fluid velocityu. The shape of the particles is constructed using a tanh-profilewith a specified radius and interface thickness and their centersof mass interact via an interparticle potential. The approach canalso be seen as a modification of a classical “model H,”29,30

with a fluid and a particle phase and the driving force in theNavier-Stokes equations governed by the interatomic poten-tial. The approach again combines continuous and discretedescriptions.

The motion of colloidal particles with positions ri(t) aregoverned by the velocities vi(t) and the evolution of a flow fieldu, where the colloidal particles are suspended in. The basicidea is to introduce concentration fields φi(r, t) ∈ [0,1] for eachparticle and to average the fluid velocity over regions with highconcentration, i.e.,

vi(t) =φi(r, t)u(r, t) drφi(r, t) dr

.

Thus, the motion of the particles can be described byri(t + ∆t) B ri(t) + ∆t · vi(t), with∆t the simulation time step.

A space-dependent fluid viscosity η f , as a function of φi,is introduced to describe the rigidity of the particles, and a

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154904-3 S. Praetorius and A. Voigt J. Chem. Phys. 142, 154904 (2015)

force term F B F[ta] to account for the particle interactions inflow equation (8). This force is chosen as the negative gradientof an interaction potential V , multiplied with the particle-concentration fields φi,

F[ta](r) def= −

i

∇rij,i

V(∥ri − r j∥)φi(r). (10)

The fluid viscosity η f = η f (1 + η) is modeled, by describingthe viscosity perturbation η, as

η(r) =i

ηp

η f− 1

φi(r), (11)

with η f < ηp the liquid and particle viscosity, respectively. InRef. 31, it is argued that the artificial diffusivity ηp/η f mustgo to∞ for the particles to become rigid. In their method, theyhave introduced a different body force to guarantee this rigiditywithout taking large values of the viscosity ratio. However, wewill here only consider the original FPD approach.

C. Towards a fully continuous description

Our aim is to derive a fully continuous model by combin-ing the FPD model with the PFC approach. A first step in thisdirection has already been done in Ref. 32, where the interac-tion potential has already been replaced by the PFC approach.The discrete off-lattice simulation for the particles is no longerneeded; the particle positions and velocities result from theadvected PFC model. However, the forcing term in the Navier-Stokes equations still requires the identification of the positionand velocity of each particle, and thus the approach still hasa discrete component. To derive a fully continuous model,we will first clarify the relation of the different approachesin Refs. 23, 28, and 32 and will show that all the discretecoupling terms can be approximated with a simple continuousexpression.

To allow for a description of the flow of individual parti-cles, we consider a variant of the PFC model, the vacancy PFCmodel, introduced in Refs. 33 and 34. Instead of minimizingthe Swift-Hohenberg functional directly, we consider a densityfield with positive density deviation ψ, only, which leads to amodification of the particle-interaction and allows to handlesingle particles, as well as many individual particles embeddedin the fluid.

II. DERIVATION OF A FULLY CONTINUOUS MODEL

In Ref. 32, the PFC model and the FPD model are com-bined, by letting the density field influence the flow field. Theinteratomic potential is encoded in Swift-Hohenberg energy(3) and the particle positions evolve according to the advectivePFC equation (see below). The forcing term F B F[ml] in theNavier-Stokes equations now ensures the fluid velocity u to beequal to the particle velocity vi at the particle position ri, i.e.,

F[ml](r) def= ω

i

(vi − u(r))δ(r − ri), (12)

with ω ≫ 1 a penalty parameter and δ(·) the pointwise delta-function. Thereby, position and velocity of each individual par-ticle must be extracted from the density field ψ by tracking the

maxima of the density that are interpreted as average particlepositions. These quantities are then explicitly inserted into theexpression of the forcing term. The fluid viscosity η f can bemodeled as before in (11), but now ψ can directly be used todistinguish between the background fluid and the particles.

Force term (12) constrains the fluid velocity to be equalto the particle velocity at the center-of-mass position of theparticle. In order to implement a no-slip boundary conditionat the particle surfaces, the delta function needs to be replacedby the concentration fields φi(r),

F[dd](r) def= ω

i

(vi − u(r))φi(r). (13)

This ansatz is highly related to the diffuse domain approach,35

where this force is shown to converge to the no-slip boundarycondition u = vi at the i-th particle surface, if the interfacewidth goes to zero. Thereby,ω has to be related to the interfacethickness, see Ref. 36 for a detailed convergence study.

In the following, we give a new formulation of a contin-uous force term that can be evaluated without extracting indi-vidual particle positions and velocities. At first, we relate thedensity field ψ, described in (7), to a delta function δ(r) and toa concentration field φ(r) =

i φi(r). In a second step, the par-ticle velocities are shown to arise directly from the evolutionequations (7) and (6), respectively.

A. Approximation of a delta-function

For the classical PFC equation in 1D, a one-mode approx-imation of the density ψ is given by Ref. 37

ψom(r) = A cos(q0r) + ψ, (14)

where A,q0, and ψ are constants that define the amplitude,lattice constant, and mean density of the field, respectively. Weintroduce

ψ(0) =12

(1 +

ψom − ψA

), ψ(k) = (ψ(k−1))2, (15)

for k > 0, or in explicit form ψ(k) = [ψ(0)]2k for k ∈ N. Afterappropriate normalization, we obtain

δ(k)(r) B Nkψ(k)(r), (16)

with Nk normalization constants that ensure the propertyδ(k)(r) dr = 1. Values for various indices k can be found in

Table I. Thus, we have a sequence of nascent delta functions.Figure 1 shows the first three elements of this sequence in com-parison with the classical Gaussians δexp

ϵ (r) e[(q0∥r∥)2/(−4ϵ)],visualizing the convergence qualitatively. As a consequenceof this property, the shifted and scaled density field ψ(0) canbe seen as a first-order approximation of a delta function. Theapproach can be generalized to 2D and 3D and will be used forψ instead of ψom.

TABLE I. The first four elements of the sequence Nk , the normalizationconstants for the nascent delta function δ(k ).

k 0 1 2 3

Nk · πq01 4

36435

16 3846 435

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154904-4 S. Praetorius and A. Voigt J. Chem. Phys. 142, 154904 (2015)

FIG. 1. The first three elements of the sequences δexpϵk (normalized), with

ϵk = 2−k , in blue (upper curves) and δ(k ) in red (lower curves). The latticeconstant is q0= 1.

B. Approximation of concentration fields

The concentration field φi in Ref. 28, used for the phase-field description of particles, is defined by

φi(r) = 12

(1 − tanh

(∥r − ri∥ − a)3ϵ

), (17)

with ri the center-of-mass position of the i-th particle, a theparticle radius and ϵ a small parameter that defines the widthof the smoothing region. We now interpret ψ(0) in (15) as aconcentration field. It has the value one at the maxima ofthe cosine profile and zero in between. The transition is verycoarse but gives an approximation of the tanh-profile of φ(r)=

i φi(r), which can be refined with

φ(ψ) = 12

(1 + tanh

(ψ(0) − σ)3ϵ

), (18)

where σ = 12

1 + cos(q0 · a) is a shifting parameter, see

Figure 2 for an example of such an implementation.In order to define the viscosity field, we adopt expression

(11) and insert for φ(r) the field φ(ψ). Thus, we introduce aviscosity field depending directly on the PFC density field ψ,using transformation (18) as

FIG. 2. Transformation of the density fieldψ into a tanh-concentration field,for different particle radii. The lattice constant is q0= 1 and interface widthϵ = 0.1.

η(r) = η(ψ(r)) =(ηp

η f− 1

)φ(ψ). (19)

C. Peak velocities

To approximate the particle velocities vi, we follow theapproach of Rauscher38 and consider a curl-free velocity fieldfor the derivation. Let u be given with the property ∇ × u = 0and ∇ · u = 0. Then, there exists a potential field Ψ such that

u = − 1γm∇Ψ, ∆Ψ = 0. (20)

Following the argumentation of Ref. 38, the flow potential Ψacts as an external potential that drives the particle density. InDDFT models, this external potential enters the free-energy byF ∗[ϱ] B FH[ϱ] + Fext[ϱ], with

Fext[ϱ] =Ψ(r)ϱ dr.

Inserting F ∗ into (2) instead of FH leads to

mϱ (∂tv + (v · ∇)v + γv)= −ϱ∇ δF∗[ϱ]δϱ

+ η∆v

= −ϱ∇ δFH[ϱ]δϱ

− ϱ∇Ψ + η∆v

= −ϱ∇ δFH[ϱ]δϱ

+ γmϱu + η∆v

and finally we arrive at

mϱ (∂tv + (v · ∇)v + γ(v − u)) = −ϱ∇ δFH[ϱ]δϱ

+ η∆v.

Going to the dimensionless form, by introducing lengthand time scales and inserting Fsh for FH gives

∂tψ + ∇ ·(1.5 + ψ)v

= 0, (21)

(1.5 + ψ) (∂tv + (v · ∇)v + Γ(v − u)) = 1Re∆v

− 1Pe

(1.5 + ψ)∇ δFsh[ψ]δψ

. (22)

In the overdamped limit, Γ ≫ 1, velocity equation (22) reducesto a simple expression for the velocity v,

Γ(v − u) ≃ − 1Pe∇ δFsh[ψ]

δψ. (23)

Inserting this into (21) results in the advected PFC equationintroduced in Ref. 23 and considered in the context of DDFTin Ref. 16,

∂tψ + u · ∇ψ = 1ΓPe∇ ·

((1.5 + ψ)∇ δFsh[ψ]

δψ

)= ∇ ·

(M(ψ)∇ δFsh[ψ]

δψ

), (24)

with a mobility function M(ψ) = 1ΓPe (1.5 + ψ).

Although this equation can only be derived for potentialflows, we will use it as an approximate model for non-potentialflows as well. With (23), we have found an explicit expressionfor the mean velocity of the particles that can be used toformulate forcing term (12) in the continuous form

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154904-5 S. Praetorius and A. Voigt J. Chem. Phys. 142, 154904 (2015)

F[ml](r) = ωi

(vi − u(r))δ(r − ri)

≈ − ω

ΓPe∇ δFsh[ψ]

δψ

i

δ(r − ri)

≈ − ω

ΓPe∇ δFsh[ψ]

δψδ(k), (25)

with δ(k) nascent delta function (16) approximating δΩ=

i δ(r − ri). The first-order approximation of this force, with

δ(0) ≈ N0ψ(0) =q0

πψ(0) =

q0

2π1 +

1A(ψ − ψ),

thus reads

F[ml](0) (r) = −(M0 + M1ψ)∇ψ(r), (26)

with M0 = (1 − ψ/A) ωΓPe

q02π , M1 =

ωΓPe

q02πA , and ψ B

δFsh[ψ]δψ

,which results in the considered fully continuous description.For ω ∼ Γ, we have M1 = O( 1

Pe ).

D. Individual particles and number of particles

In order to allow for particles to move freely, we add amodification introduced by Ref. 33. The authors have arguedthat by limiting the field ψ from below, the particle interactioncan be modified. Therefore, they have introduced the constraintψ ≥ 0, which allows to control the volume fraction of particlesin the domain by changing the mean density of the system.

To implement the constraint, the free-energy is modifiedby including a penalty term, i.e., Fvpfc B Fsh + Fpenalty, with

Fpenalty[ψ] =

β(|ψ |n − ψn) dr,

with β ≫ 1 and n an odd integer exponent.The variational derivative of Fpenalty can be found to be

b(ψ) B δFpenalty[ψ]δψ

= nβψn−1(sign(ψ) − 1), (27)

with sign(ψ) =

1, for ψ > 0,0, for ψ = 0,−1, for ψ < 0.

While localized states are also observed in the originalPFC model for a small range of parameters in the coexistenceregime,39 we consider the approach in Refs. 33, 34, and 40using penalty term (27). Here, the number of particles canbe controlled by choosing the mean density and the area theparticles occupy. The initial density field for a collection ofN particles located at the positions ri, i = 1, . . . , N , is acomposition of local density peaks,

ψ(i)0 (r) =

A ·cos(√

3q0

2∥r − ri∥) + 1

for ∥r − ri∥ < d

20 otherwise,

summed up to ψ(r) = Ni=1ψ

(i)0 (r).

Thus, each particle occupies an area of approximatelyBp B π(d/2)2 in 2D. Based on the ideas in Ref. 33, we set themean density in the particle domain toψ1=

(−48 − 56r)/133,as well as r = −0.9 and q0 = 1. The last two parameters definethe mean density of the system as

ψ =N · Bp

B0ψ1,

with B0 = |Ω| the area of the computational domainΩ and withA = ψ1 the parameter for the density scaling.

E. Navier-Stokes PFC model

Combining all the components, i.e., the Navier-Stokesequation for solvent (8) with viscosity η given by η(ψ) in (19),and volume force F by expression (26), combined with densityevolution (24) with Fsh or Fvpfc, results in the fully continuousNavier-Stokes PFC (NS-PFC) model

∂tu + (u · ∇)u = ∇ · σ − M1ψ∇ψ,∇ · u = 0,

∂tψ + u · ∇ψ = ∇ ·M(ψ)∇ψ,

ψ =δFsh/vpfc[ψ]

δψ,

(28)

with

σ = −pI +1

Re f

1 + η(ψ)

∇u + ∇u⊤,

δFsh[ψ]δψ

= ψ3 + (r + (1 + ∆)2)ψ,δFvpfc[ψ]

δψ= ψ3 + (r + (1 + ∆)2)ψ + b(ψ),

and p = p + M0ψ a rescaled pressure. Besides the definition

of ψ, these equations have exactly the form of “model H”as considered in Ref. 41. In Appendix B, we demonstratethermodynamic consistency of the derived model.

III. NUMERICAL STUDIES

We now turn to quantitative properties of the model andcompare it with the original PFC model and the FPD approachof Ref. 28 for various situations. We rewrite the NS-PFCsystem as a system of second order equations. Therefore, thevariational derivatives are implemented as

δFsh[ψ]δψ

= ψ3 + (r + 1)ψ + 2∆ψ + ∆ν,

δFvpfc[ψ]δψ

= ψ3 + (r + 1)ψ + 2∆ψ + ∆ν + b(ψ),ν = ∆ψ.

System (28) has to be solved for u, p,ψ,ψ,and ν in a domainΩwith boundary conditions depending on the specific example.To numerically solve this system of partial differential equa-tions, we apply here an operator splitting approach42 with asequential splitting, where we solve the PFC equations first,followed by the Navier-Stokes equations. In time, we use asemi-implicit backward Euler discretization with a lineariza-tion of all nonlinear terms, i.e., a one-step Newton iteration.In space, we discretize using a finite element method, withLagrange elements, e.g., a P2/P1 Taylor-Hood element for theNavier-Stokes equation and a P2 element forψ,ψ,and ν in thePFC equation. We further use adaptive mesh refinement, lead-ing to an enhanced resolution along the particles. The systemis solved using the parallel adaptive finite element frameworkAMDiS.43,44

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154904-6 S. Praetorius and A. Voigt J. Chem. Phys. 142, 154904 (2015)

A. Crystallization

The first numerical examples use Fsh and consider crystal-lization processes in flowing environments. The fluid is drivenby boundary conditions. In the first case, we consider a rotatingfluid, i.e., a gyre flow and in the second case, a Poiseuille flowwith a parabolic inflow velocity profile.

1. Rotating crystals

A crystal grain is placed in a rotating fluid initially givenby

∂tr = u0(x, y) = *..,

C sin(π xdimx

) cos(π y

dimy)

−C cos(π xdimx

) sin(π y

dimy)

+//-

(29)

in a domain (x, y) ∈ Ω = [0,dimx] × [0,dimy]. For the numer-ical experiment, we have chosen dimx = dimy = 42d. Theboundary conditions for the Navier-Stokes equations are setby u0.

We start the growth process with an initial grain of radius2d in an undercooled environment with parameters r = −0.3and mean density ψ = −0.35. The mobility function is set toM(ψ) = ψ + 1.5 and the force scaling to M1 = 1. The fluidReynolds number is set to Re f = 1 and the viscosity ratio toηp/η f = 100. For the concentration field that defines the profileof the viscosity, we have used an approximation of ψ(0),

φ B ψ(0) ≈ψ −minΩ(ψ)

maxΩ(ψ) −minΩ(ψ) .Thus, the fluid viscosity is high in particles, low in betweenparticles, and takes an intermediate value in the isotropic phaseaway from the crystal. However, other definitions of the viscos-ity parameter are possible as well.

In Fig. 3, the growth shapes for different velocities C areshown at the same simulation time. For a still fluid (C = 0),i.e., no advection, the final shape is the largest and the sizeof the crystal decreases for increasing velocity. For the largestconsidered velocity C = 1, the faceting of the crystal is alsomore pronounced than for the case of no induced fluid flow.The stationary images also show that the crystal rotates duringthe growth process. This can be seen at the varying crystalorientations in (a), (b), and (c) indicated by the white angle.

The growth process is analyzed in Fig. 4, showing theradius of the growing crystal over simulation time. The growthvelocity strongly depends on the induced fluid velocity, asshown in the inlet plot of Fig. 4. The crystal grows slower for a

FIG. 3. Final growth-shapes of the crystal in a flowing environment at timet = 3000. Shown is the particle density ψ with color red corresponds to highdensity and blue to low density. The fluid velocity denoted by C : (a) C = 0,(b) C = 0.5, and (c) C = 1. The white angles show the crystal orientation andthus give an indication for crystal rotation.

FIG. 4. Radius of the crystal divided by lattice constant over dimensionlesstime, for various fluid velocities C . In the inlet, the growth velocity of thecrystal normalized by the lattice constant Vg is shown for the final timet = 3000.

larger induced fluid velocity. This clearly shows one directionof the coupling, i.e., the fluid influences the crystallization.

The opposite can be found as well. The crystal alsochanges the velocity profile of the fluid. In the inlet of Fig. 5,the velocity profiles of two fluids are compared. The left imageshows the profile of a fluid with no backcoupling of the densityfield to the Navier-Stokes equations. This essentially shows theinitial profile u0. The right image shows the velocity profilefor the full NS-PFC model with C = 1. We observe differentmagnitudes of the velocity, whereas the streamlines do notchange qualitatively. A more detailed analysis of the velocityprofile along the x-axis from the center to the boundary ofthe domain can be found in Fig. 5. With fluid coupling, alinear increase of the magnitude in the domain of the crystalis observed, indicated by the black dashed line, which is

FIG. 5. Fluid velocity at time t = 3000 extracted from positive x-axis, asindicated by the lines in the inlet pictures. The slope 0.04 corresponds to theangular velocity of the fluid in the region with a radius less than the crystalradius. In the inlet, the magnitude of fluid velocity in the domain Ω for C = 1is shown. Left: fluid flow not influenced by the crystal. Right: crystal slowsdown the fluid due to higher viscosity in the region of particles.

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154904-7 S. Praetorius and A. Voigt J. Chem. Phys. 142, 154904 (2015)

FIG. 6. Crystal shape and corresponding velocity profile at time t= 3800in a narrow channel. Left: density field ψ, with arrows in the lower halfcorresponding to u−vcrystal, i.e., fluid velocity relative to the translationvelocity of the crystal. Right: fluid velocity u with contour line that indicatesthe shape of the crystal. In the lower half, the velocity relative to the channelflow velocity, i.e., u−u0, is shown. Color red corresponds to high values andblue to low values.

lower than the prescribed initial profile. The crystal acts as arotating solid in the fluid, with normalized angular velocityω = ∥u∥/(∥r∥/d) = 0.04. Away from the crystal, the velocityincreases up to the prescribed boundary velocity C.

2. Translating crystals

In the second case, the crystal grows in a Poiseuille flow. Ina narrow channel, we enforce a parabolic velocity at the inflowboundary, i.e.,

u0(x, y) = (4C y(1 − y),0)⊤, y B y

dimy,

with a maximal inflow velocity C and a top/bottom boundaryvelocity set to zero. Again, we start with an initial grain ofradius 2d in the center of a box Ω with dimensions dimx

= 168d and dimy = 42d. The simulation parameters are thesame as above in the case of a rotating fluid.

The shape of the growing crystal is influenced by thefluid, which induces an anisotropy. This can be seen in Fig. 6,where the shape corresponding to a fluid velocity C = 0.15 isshown in a clipping of the whole domainΩ. The flow directionis from left to right. The particle density ψ is shown in theleft image together with the velocity relative to the velocityof the translating crystal, i.e., vcrystal = (vcrystal,0)⊤ with vcrystal≈ 0.124. The right image shows in the upper half, the absolutevalue of the velocity, with a constant value within the crystal,and in the lower half, the flow velocity relative to the initialvelocity u0. This shows an elongated vortex. In the case ofno fluid coupling, the crystal grows isotropically to a circularshape, as in the example above.

Thus, also for Poiseuille flow, we see a coupling in bothdirections, the shape of the crystal is influenced by the flowingenvironment, and the fluid velocity is influenced by the crystal.

B. Sedimentation

In the following, we apply the NS-PFC model to a collec-tion of individual particles to show the applicability as a modelfor particle dynamics. We therefore consider Fvpfc. For penaltyterm (27), we use the parameters (n, β) = (3,2000) in all ofthe following simulations. The Reynolds number and viscosityratio are chosen as before, but the viscosity profile is now

given by

φ B ψ(0) ≈ψ

maxΩ(ψ) .Thus, we have the lower fluid viscosity away from the particlesand a high viscosity on the particles. In order to stabilize theshape of the particles, we increase the diffusional part, i.e., thePeclet number Pe, respective to the mobility function M(ψ).We have chosen M(ψ) ≡ 16 in the following examples.

1. One spherical particle in a confinement

The objective of this study is to calculate the positionand velocity of one spherical particle (circular disk) settlingdown in an enclosure due to a gravitational force g. In orderto include this force, we use a Bousinesq approximation andadd the forcing term Fg B φ(ψ)g to Navier-Stokes equationsin (28).

The box dimensions are chosen to be multiples of theparticle size. All lengths are again normalized by the particleinteraction distance d = 4π/

√3, i.e., the lattice constant. We

consider the following boundary conditions:

with nΣ the outer normal to Σ B ∂Ω.Due to the symmetry of the system, we expect a symmetric

trajectory, a straight line in the center of the box with theparticle slowing down at the bottom. Fig. 7 shows the y(t)component of the evolution curve

x(t), y(t) compared with

FPD simulations. We further show the comparison of the veloc-ity profiles. For both criteria, we obtain an excellent agreement.

In the FPD setup, we have used the normalized densityfield ψ(0)(r) as a concentration field instead of tanh-profile (17)and for treatment of the wall-boundary we have introduced a

FIG. 7. Trajectory and velocity of one particle settling down in a box filledwith a liquid with fluid viscosity η f = 0.1, particle viscosity ηp = 10, andgravitational force g= (0,−1). Left: vertical position of the particle, start-ing from an initial height of 10d. Right: effective velocity of the particle,i.e., v1(t)2= (x(t)2+ y(t)2)/d2.

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154904-8 S. Praetorius and A. Voigt J. Chem. Phys. 142, 154904 (2015)

FIG. 8. Two particles settling down in an enclosure. Fluid viscosity is set toη f = 1 (a) and (b) and η f = 0.1 (c,d), particle viscosity to ηp = 100 (a,b), andηp = 10 (c) and (d) and mobility of the NS-PFC-model to M (ψ)≡ 2. Left:trajectories of the particles with coordinates ri(t)= (xi(t), yi(t)), i = 1,2.Right: absolute velocities: vi(t)2= (xi(t)2+ yi(t)2)/d2, i = 1,2.

repulsive potential

VB(k,p)(r) B kd−1distΣ(r)p,

with k = 1,p = 20, and distΣ(r) the distance of r to the bound-ary Σ of the domain Ω.

Further care is needed in order to guarantee a symmet-ric solution. Within both approaches, we use a symmetrictriangulation of the domain and symmetric quadrature rules.Otherwise, we get symmetry breaking in the trajectories, sincethe motion on a straight line is unstable with respect to smallperturbations, as it is also pointed out in the work of Ref. 24.

2. Two interacting particles

For two particles sedimenting in a box, additional hydro-dynamic interactions are expected to influence the motion ofthe particles. We expect to see the phenomena of trailing,drafting, kissing, and tumbling of the particles, as found inexperimental studies45 and as also observed in several numer-ical studies with various methods, e.g., Refs. 24, 46, and 47.Again, we compare it with FPD simulations where we haveto apply direct particle-particle interaction potentials, defined

asV(r) B k r

d

p1 − 2 rd

p2, with (k,p1,p2) = (1,12,6) and a

boundary interaction potential VB as above. Since we do nothave a one-to-one mapping between these potentials and theirrepresentation in Fvpfc and since the PFC-model introducesadditional diffusion due to a non-vanishing mobility functionM(ψ), equality of particle trajectories and particle velocitiescannot be expected. However, the results qualitatively agree fordifferent fluid viscosities, as can be seen in Fig. 8. To analyzethe dependency of the trajectories on the considered interactionpotential V , FPD simulations with different potentials, i.e.,different parameters in the Lennard-Jones type interaction,and purely repulsive interactions are performed and comparedwith each other. The obtained differences in the trajectoriesand the particle velocities are in the same order as the differ-ences if compared with the NS-PFC simulations (results notshown).

The system considered here consists of two particlesplaced below each other with a small (symmetric) displace-ment relative to the middle vertical axis. The initial configu-ration is chosen as r1 = (−0.1d,9d) and r2 = (0.1d,10d), withboundary conditions as for the case of one particle. Comparedto the one-particle case, the box size is chosen wider, i.e., awidth of 18d instead of 8d, to further reduce boundary effects.

The solution can also be compared qualitatively to theresults in Refs. 24 and 48, where the authors have studiedthe sedimentation of two hard sphere particles in a narrowenclosure in a similar setup and they have found similar trailingand drafting phenomena. However, they are not as strong asin the FPD or in our simulations. The particles start in closecontact and accelerate up to a critical time, when they startmoving apart from each other. In the visualized scenarios inFig. 8, the particle behind overtakes the other one and reachesthe bottom first. Compared with FPD in our simulations, theparticles move further away from each other and the velocitydecreases in a similar way up to the contact with the lowerboundary.

3. Many particles in an enclosure

With three particles already, the interaction and motionof the particles becomes chaotic, as pointed out in Ref. 49and discussed in detail in the review.50 Therefore, a directcomparison of trajectories is no longer meaningful. However,considering not only a few but also a larger number of particlesin a bounded box under gravity cause new effects. Particlesdo not settle down homogeneously; their dynamics stronglydepend on the distance to the walls. During the sedimentation

FIG. 9. Four snapshots of the sedimentation simulation for 120 particles in a square box. Color red corresponds to high absolute velocity and blue to lowvelocity. Left: initial configuration of particles. Second image: an instability of the particle front, starting from the boundaries. Third image: particles start tosediment at the bottom. Right: final compressed sediment of particles.

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154904-9 S. Praetorius and A. Voigt J. Chem. Phys. 142, 154904 (2015)

FIG. 10. Evolution of mean density of particles for four different time stepscorresponding to the snapshots in Fig. 9. The final configuration of particlesin a hexagonal lattice has a higher density than the initial configuration in asquare lattice.

process, Rayleigh-Taylor-like instabilities and fingering occurand a compression of the particle lattice at the bottom of thebox is seen. To demonstrate the possibility of our approach, todeal with moderate numbers of particles, we aim to observethese phenomena. We have studied a situation of 120 particlesarranged in a square lattice in the upper part of a squaredomain. The initial distance of neighboring particles is set tothe lattice constant d. The width of the box is chosen so that20 particles fit perfectly in one horizontal line, i.e., we havedimx = dimy = 20d. Boundary conditions are chosen similarto the cases before. For a gravitational force g = (0,−2)⊤ wehave simulated the sedimentation process in a fluid with aviscosity ratio ηp/η f = 100, as above. The particles close tothe side walls start settling down first and due to their motion,an upwards fluid flow in the center of the domain is induced. Avisualization of the sedimentation process is shown in Fig. 9.We have drawn black circular disks to indicate the particle posi-tions. Four snapshots are shown, the initial and final configu-ration and two intermediate states, i.e., the beginning of thedevelopment of the instability and a snapshot with partiallysedimented particles.

In Fig. 10, the mean particle concentration ⟨ψ⟩(y) isshown, which is obtained by averaging over stripes of widthd along the particle layers,

⟨ψ⟩(y) B xmax

x=xmin

y+0.5d

y′=y−0.5dψ(x, y ′) dy ′dx.

The high-concentration region moves from top to bottom overtime and the mean particle density is higher at the bottom ofthe box than for the initial configuration.

IV. CONCLUSION

A fully continuous model is developed to simulate col-loidal particles in a fluid, interacting via direct particle-particleinteraction and via the induced flow fields. The method isbased on ideas of dynamic density functional theory and fullyresolved direct numerical simulations. The derived NS-PFC

system operates on diffusive time scales and provides a quali-tative approach down to the particle size.

We have demonstrated the quality of the method in variousexamples; first, in crystallization processes analyzing the influ-ence of a macroscopic flow field and second, for three commontest cases, namely, the sedimentation of one, two, and manyparticles. For one and two particles, we have quantitativelycompared the trajectories and velocities obtained by our simu-lation to simulations with the FPD method and have foundgood agreement. For the case of many particles, we see theexpected instabilities and compression at the bottom. For morequantitative comparisons, the calculation of a hydrodynamicradius would be desirable, as it is done by Refs. 3 and 12.However, this requires computations in 3D for the comparisonwith Stokes drag and drag torque and has thus to be consideredin future work. As a preliminary step, we have computed theeffective radius of the vacancy NS-PFC model by comparisonwith the FPD method, using concentration profile (17), forwhich a radius is given. We have computed the particle veloc-ities obtained by both methods for one particle dragged througha periodic channel. Comparing the result for various radii forthe FPD method, we find an effective radius for the NS-PFCmodel of approximately 0.35d.

However, in comparison with the other methods men-tioned in the Introduction, we do not see the advantages of ourapproach in a more detailed description of the underlying phys-ics on a single particle scale, but in the emergent phenomena onlarger scales. Here, the formulation as a fully continuous modelhas several numerical advantages: we expect stable numer-ical behavior. For the classical PFC equation time step, inde-pendent stability can be proven for the discrete scheme.51–53

Coupling this to the Navier-Stokes equation, as considered,e.g., in Ref. 54, allows for larger time steps as in the explicitcoupling schemes of hybrid methods. As the NS-PFC modelonly contains local terms, the algorithms are expected to scaleindependently of the number of particles. Numerical detailsof an efficient parallel scheme are described in Ref. 53 forthe classical PFC equation and can be extended towards thecoupling with Navier-Stokes.

ACKNOWLEDGMENTS

This work has been funded through Grant Nos. DFGVo899/11 and FP7 IRSES 247504. We further acknowledgethe provided computing resources at ZIH at TU Dresden andJSC at FZ Jülich.

APPENDIX A: DIMENSIONLESS FORM

The density ϱ is driven by the variational derivative of theHelmholtz free energy FH. This functional can be decomposedinto two contributionsFH = Fid + Fexc, where the ideal gas partFid is known and the excess free part unknown for generalsystems,

FH[ϱ] = kBT

ϱ[ln(Λdϱ) − 1]dr + Fexc[ϱ],with kB Boltzmann’s constant, T the temperature, Λ the ther-mal de-Broglie wave-length, and d the space dimension.

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154904-10 S. Praetorius and A. Voigt J. Chem. Phys. 142, 154904 (2015)

Inserting a parametrization ϱ = ϱ(ϕ) = ϱ(1 + ϕ) with thedensity deviation ϕ and reference density ϱ into the energyand expanding the ideal gas part of the energy around ϱ leadsto a polynomial form of Fid. Using a Ramakrishnan-Yussouffapproximation55 of the excess free part results in an expressionof the two-point correlation function c(2)(r,r′, ϱ),Fexc = C +

(ϱ(r) − ϱL)c(2)(r,r′, ϱL)(ϱ(r′) − ϱL)drdr′,

with ϱL a reference liquid density. This corresponds to a convo-lution of (ϱ − ϱL) with c(2) and can thus be transformed intoa product in Fourier space. Expanding c(2) around the wave-number zero and transforming back lead to a gradient expan-sion of Fexc that can be written in the variable ϕ,

1kBT ϱ

(FH[ϱ(ϕ)] − FH) ≈

12ϕ2 − 1

6ϕ3 +

112ϕ4dr

−

12ϕ(C0 − C2∆ + C4∆

2)ϕdr

+

D0 + D1ϕdr,

with C0,C2,C4,D0,D1 expansion coefficients. For a detailedderivation, see, e.g., Refs. 21–23. Since we take the gradient ofthe variational derivative in the dynamical equations, all con-stant and linear terms can be neglected in the energy withoutchanging the dynamics.

Fixing the lattice spacing L, the dimensionless bulk modu-lus of the crystal B and introducing parameters r and ψ0, with

L2B2|C4|C2

, BBC2

2

4|C4| =ψ2

0

3,

sign(C4)= −1, rB ψ−20

(94− 3C0

)− 1,

scaling the length by L, i.e., r = r(r) B rL

, and introducing thederivatives ∇ B ∂r, ∆ = ∇ · ∇ and a new variable ψ = ψ(r) as

ϱ(r) = ϱ(ψ0 · (ψ r)(r) + 1.5),with (ψ r)(r) = ψ(r(r)), where acts as a function composi-tion operator, result in the classical PFC energy

3kBT ϱψ4

0

(FH[ϱ(ψ)] − FH) ≈ Ld

12(1 + r)ψ2 +

14ψ4

+ψ∆ψ +12ψ∆2ψ dr

C LdFsh[ψ].We consider the variational derivative of FH and relate it

to the variational derivative of Fsh,

δFH[ϱ]δϱ

=1

Ld

(δFH[ϱ r]δ(ϱ r) r

)≈ 1

Ld*,

δ 1

3 kBT ϱψ40LdFsh[ψ] + FH

δ(ϱ r) r+-

=13

kBT ϱψ40

(δFsh[ψ]δψ

δψ

δ(ϱ r) r)

=13

kBTψ30

(δFsh[ψ]δψ

r).

Inserting the parametrization of ϱ into dynamical equations(1) and (2), fixing ψ0 = 1 for simplicity and using the length

scaling r give

(1.5 + ψ)∂tv + 1L(v · ∇)v + γv

=

kBT3mL

∇ δFsh[ψ]δψ

+η

m ϱL2 ∆v,

∂tψ = −1L∇ ·

(1.5 + ψ)v.

Introducing the dimensionless variables t B tV0/L and v B v/V0 finally gives the dimensionless dynamical equations

(1.5 + ψ)∂tv + (v · ∇)v + γLV0

v=

kBT3mV 2

0

∇ δFsh

δψ

+η

m ϱLV0∆v,

∂tψ = −∇ ·(1.5 + ψ)v

.

By defining the dimensionless numbers

Pe =3mV 2

0

kBT, Re =

m ϱLV0

ηand Γ =

γLV0

as above, we find Eqs. (4)–(5), where we have neglected thehat symbol on the derivatives for readability.

APPENDIX B: ENERGY DISSIPATION

To demonstrate thermodynamic consistency of the model,we assume that the total energy of the system is composedof the Helmholtz free-energy FH, respective to an appropriateapproximation of this functional and the kinetic energy

Fkin =ρ f

2

∥u∥2 dr

of the surrounding fluid. To be consistent with dynamicalequations (28), we focus on the dimensionless energies byintroducing length and time scales as above and by definingdimensionless variables denoted by a hat symbol. Additionally,we normalize the energies

r = r/L, t = tV0/L, u = u/V0, F∗ = F∗/(V0L2η f ).This gives us the dimensionless kinetic energy

Fkin =Re f

2

∥u∥2 dr

and by considering the correct scaling of the Swift-Hohenbergenergy FH ≈ kBT ϱ L3

3 Fsh (see Appendix A), we find

FH =1Sc

14ψ4 +

12ψ

r + (q2

0 + ∆)2ψ dr,

with the Schmidt number Sc given by

Sc =Pe

Re f

¯ϱ, with ¯ϱ Bρ f

m ϱ.

The total dimensionless energy, to be considered, nowreads

Ftot = Fkin + FH.

In the following, we consider only nondimensional variablesand for readability drop the hat symbols.

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154904-11 S. Praetorius and A. Voigt J. Chem. Phys. 142, 154904 (2015)

We assume that the evolution equations for momentumand mass conservation read

∂tu + (u · ∇)u = ∇ · σ + F,∇ · u = 0,

∂tψ + u · ∇ψ = −∇ · j,(B1)

where the volume force F and the flux j need to be determinedto justify thermodynamic consistency. LetΩ be a fixed domainwith Lipschitz-boundary Σ. The time-evolution of the energyFtot can be split into

Fkin = Re f

Ω

u · ∂tu dr

= Re f

Ω

u · (−(u · ∇)u + ∇ · σ + F) dr,

FH =1Sc

Ω

δFsh[ψ]δψ

∂tψ dr.

(B2)

Using incompressibility and integration by parts and the rela-tions

12∇

∥u∥2= (u · ∇)u − (∇ × u) × u,

f ∇u, D(u)

Ω=

f D(u), D(u)

Ω,

for a scalar field f = f (r) and the inner product (A,B)Ω=Ω

A : B dr, we getΩ

u · (u · ∇)u=Ω

12

u · ∇∥u∥2

+ u ·(∇ × u) × u

dr

=12

Σ

(u · nΣ)∥u∥2 dΣ

= 0 for

u · nΣ = 0, (no-penetration),

u = 0, (no-slip),Ω

u · ∇ · σ dr=Ω

−∇u : σ dr ≤0

+

Σ

u · σ · nΣ dΣ (∗)

(∗)= 0 for

σ · nΣ = 0, (no-flux),

u = 0, (no-slip).

Thus, we get for the kinetic part of the energy, in case of no-slipboundary conditions, the estimate

Fkin ≤ Re f

Ω

u · F dr.

The derivative of the PFC-part of the energy evolution reads

FH =1Sc

Ω

j∇ δFsh[ψ]δψ

− u · δFsh[ψ]δψ

∇ψ dr.

By choosing the flux j proportional to −∇δFsh[ψ]/δψ, e.g.,

j = −M(ψ)∇ δFsh[ψ]δψ

,

with M(ψ) any positive definite function, we find for the totalenergy evolution

Ftot ≤Ω

u ·Re fF −

1ScδFsh[ψ]δψ

∇ψ

dr

and can choose F so that this integral vanishes, i.e.,

F =1

Re fScδFsh[ψ]δψ

∇ψ =¯ϱ

PeδFsh[ψ]δψ

∇ψ.

Using incompressibility again, we get the relation to theforce and flux terms derived before. For no-slip boundaryconditions, we have

Ω

− 1Sc

u · δFsh[ψ]δψ

∇ψ dr =Ω

1Sc

u · ψ∇ δFsh[ψ]δψ

dr

and thus, the force

F = −¯ϱ

Peψ∇ δFsh[ψ]

δψ(B3)

and with M1 = ¯ϱ/Pe, the above set of Eq. (28). Our derivedcontinuum model thus fulfills thermodynamic consistency.

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